﻿12 
  Dr. 
  J. 
  \V. 
  Nicholson 
  on 
  the 
  Relation 
  oj 
  

  

  whence 
  

  

  J 
  

  

  div(cos 
  w^ 
  4- 
  sin 
  ^v^ 
  + 
  ^-«'^)e-*""' 
  

  

  

  

  = 
  „-^i(V3 
  + 
  l){J-i-Ji} 
  . 
  . 
  . 
  (18) 
  

   and 
  

  

  1 
  c?io(cos 
  iv^ 
  - 
  sin 
  w;3)e-^''' 
  - 
  | 
  dw 
  , 
  ^-«Hor*f 
  

   Jo 
  Jo 
  

  

  By 
  addition 
  and 
  subtraction, 
  we 
  may 
  express 
  the 
  integrals 
  

  

  die 
  . 
  (cos 
  16"^. 
  6"'^"'— 
  sinh 
  aw 
  . 
  e''"^') 
  

  

  J. 
  

  

  /^ 
  00 
  

  

  I 
  (/?o(sin 
  10 
  

   Jo 
  

  

  Again, 
  another 
  particular 
  integral 
  o£ 
  

  

  (20) 
  

  

  ;3^-'^""+ 
  cosho-it'.<?-''^) 
  

  

  

  IS 
  

  

  ^=1 
  ^i?iy 
  sin 
  (ic^ 
  — 
  oiv). 
  

   Jo 
  

  

  With 
  the 
  aid 
  of 
  this, 
  we 
  may 
  express 
  the 
  two 
  integrals, 
  

  

  /loo 
  

  

  I 
  die 
  (^~<^"' 
  — 
  sin 
  criv) 
  cos 
  iv^ 
  1 
  

  

  •'" 
  _ 
  in) 
  

  

  dw 
  (6~^^ 
  cos 
  crio) 
  sin 
  lo^ 
  \ 
  

  

  J, 
  

  

  It 
  may 
  be 
  shown 
  that 
  no 
  other 
  integrals 
  of 
  the 
  linear-cubic 
  

   type 
  can 
  be 
  so 
  expressed 
  unless 
  they 
  are 
  derived 
  by 
  the 
  

   processes 
  of 
  addition 
  and 
  subtraction 
  from 
  those 
  already 
  

   given. 
  

  

  Asymptotic 
  expansions. 
  

  

  In 
  a 
  previous 
  paper,* 
  the 
  author 
  has 
  shown 
  that 
  when 
  

   n—z 
  is 
  not 
  very 
  great 
  in 
  comparison 
  with 
  ^a, 
  which 
  is 
  itself 
  

  

  * 
  Phil. 
  Mag. 
  August 
  1008. 
  

  

  